Hi,
we will give to all the polinomials formulas that
give the primitive of Pell's equation x^2-Ay^2=1
*******************************************
(1)
If A=n^2-n, then y=2 and x=2n-1 (with n>=2)
A=(2,6,12,20,30,42,56,72,90,110,....,)
Y=(2,2,2,2,2,2,2,2,2,2,...,)
X=(3,5,7,9,11,13,15,17,19,21,...)
*******************************************
(2)
If A=n^2-2n, then y=1 and x=n-1 (with n>=3)
A=(3,8,15,24,35,48,63,80,99,120,...,)
Y=(1,1,1,1,1,1,1,1,1,1,...,)
X=(2,3,4,5,6,7,8,9,10,11,...,)
*******************************************
(3)
If A=n^2+1, then y=2n and x=2n^2+1 (with n>=1)
A=(2,5,10,17,26,37,50,65,82,101,...,)
Y=(2,4,6,8,10,12,14,16,18,20,...,)
X=(3,9,19,33,51,73,99,129,163,201,...,)
********************************************
(4)
If A=n^2-2, then y=n and x=n^2-1 (with n>=2)
A=(2,7,14,23,34,47,62,79,98,119,...,)
Y=(2,3,4,5,6,7,8,9,10,11,...,)
X=(3,8,15,24,35,48,63,80,99,120,...,)
********************************************
(5)
If A=n^2+2, then y=n and x=n^2+1 (with n>=1)
A=(3,6,11,18,27,38,51,66,83,102,...,)
Y=(1,2,3,4,5,6,7,8,9,10,...,)
X=(2,5,10,17,26,37,50,65,82,101,...)
********************************************
(6)
If A=(n^2+1)/25, then y=10n, and x=2n^2+1
(with n>7, and n^2+1=0 mod.25
n=(18,32,43,57,68,82,...,)
A=(13,41,74,130,185,269,...,)
Y=(180,320,430,570,680,820,...,)
X=(649,2049,3699,6499,9249,13449,...,)
*********************************************
continued...
Regards
Vincenzo Librandi
vincenzo.librandi@tin.it
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